When rotating an object, the rotation degree increments depend on several things.
1) If the grid is on, the rotation conforms to the grid. Increasing the rulers divisions will allow you to rotate in finer increments. Likewise, turning the grid OFF will allow you to rotate in pixel-level increments.
2) How far you drag from the pivot point. After selecting the objects handle to rotate, you can drag away from the object while rotating. This will give you a wider "sweep" or circumference around the pivot point of rotation. A wider sweep will let you rotate in finer increments.
3) The zoom level. If you zoom out to a reduced view, then drag away from the pivot point while rotating, you'll get an even wider sweep.
To get the maximum precision in rotating, turn the grid off, zoom out to a reduced view, then drag away from the object's handle while rotating.